Vietoris--Rips complexes of ellipses at larger scales

TL;DR

This paper explores the homotopy types of Vietoris--Rips complexes of ellipses at larger scales, identifying specific scale parameters where these complexes are homotopy equivalent to spheres or wedge sums of spheres.

Contribution

It extends previous work by characterizing the homotopy types of Vietoris--Rips complexes of ellipses at larger scales, revealing new topological equivalences based on scale parameters.

Findings

Vietoris--Rips complexes of ellipses can be homotopy equivalent to a 3-sphere at certain scales.

They can also be homotopy equivalent to a wedge sum of 4-spheres or a 5-sphere at other scales.

The paper provides explicit scale parameters as functions of eccentricity for these homotopy types.

Abstract

For $X$ a metric space and $r>0$, the Vietoris--Rips simplicial complex $\mathrm{VR}(X;r)$ has $X$ as its vertex set, and a finite subset $\sigma \subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than $r$. In ``On Vietoris--Rips complexes of ellipses'', the authors studied the homotopy types of Vietoris--Rips complexes of ellipses $E_a=\{(x,y)\in \mathbb{R}^2~|~(x/a)^2+y^2=1\}$ of small eccentricity, meaning $1<a< \sqrt{2}$, at small scales $r < \frac{4\sqrt{3}a^2}{3a^2+1}$. In this paper, we further investigate the homotopy types that appear at larger scales. In particular, we identify the scale parameters $r$, as a function of the eccentricity $a$, for which the Vietoris--Rips complex $\mathrm{VR}(E_a;r)$ is homotopy equivalent to a $3$-sphere, to a wedge sum of $4$-spheres, or to a $5$-sphere.